Question

Today we talked about the Doppler Effect...how an observer can observe a higher or lower frequency than a source is actually emitting. This effect can happen if the observer and/or the source are moving towards/away from each other.

We derived the equation that governs how the observed frequency (fo) relates to the regular frequency of the source (fs). That equation is (fo)=(fs)*[v +/- vo]/[v +/- vs]. Your job in this essay is to explain how you know to choose the plus or the minus in the numerator and denominator. Pick at least two sample cases and for each sample case include both a physical reason you'd expect fo to go up or down (think about how sound waves are squishing together or spreading apart) and a mathematical reason (like will picking a + in the denominator increase or decrease fo?)

Answer 1

The direction of movement in the Doppler effect dictates the use of plus or minus signs in the formula. For a source moving towards a stationary observer or an observer moving towards a stationary source, the observed frequency increases. This is due to sound waves appearing compressed, leading to a higher perceived frequency.

Step-by-step explanation:

The Doppler effect governs the observed frequency (fo) of a sound wave as experienced by a stationary observer when the source is moving towards or away from the observer. When selecting the plus or minus in the Doppler effect formula fo = fs(v ± vo)/(v ± vs), it depends on the direction of the motion in relation to the observer.

If the source is moving towards the observer, the frequency increases; thus, in the numerator, we use a minus sign (since the approach is considered a negative velocity relative to the observer) and in the denominator, if the source is moving towards the observer, we also use a minus sign (which results in a smaller denominator and thus a larger fo).

For the first sample case, consider a source moving towards a stationary observer. Physically, as the source approaches, the sound waves seem to compress, leading to an increased frequency, which is why we deduct the source's velocity (vs) in the formula's denominator, mathematically resulting in a higher fo.

For the second case, when the observer moves towards a stationary source, the observer encounters sound waves more frequently, leading to a higher perceived frequency. Mathematically, adding the velocity of the observer (vo) in the numerator will increase the fo, matching the physical intuition.